Acoustics & Music Theory
why it sounds like that
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The comma that will not close

Twelve fifths do not equal seven octaves. Every tuning system in history is a different decision about where to hide the difference.
CC♯DE♭EFF♯GA♭AB♭Bstartseven octavestwelve fifthsthey miss by 23.46 centsone turn is one octave · angle is pitch class · radius is how far it has climbed Pitch and tuning

Twelve fifths and seven octaves, which are not the same thing

Stack twelve perfect fifths and the note that arrives should be the one seven octaves up. It is sharp by about a quarter of a semitone, and the whole history of tuning is a set of decisions about what to do with that.

 024681012-20-1001020semitones above the tonicCC♯DE♭EFF♯GA♭AB♭BCjust intonationPythagoreanquarter-comma meantonesharp of equalflat of equal Pitch and tuning

Where to hide the comma, which is the only real question

Four tuning systems, four different places to put an error that cannot be removed. Each one is coherent, each was standard somewhere, and the choice between them decided what music could be written.

 E♭B♭FCGDAEBF♯C♯G♯D♯-5.4-5.4-5.4-5.4-5.4-5.4-5.4-5.4-5.4-5.4-5.4+35.7each fifth's departure from a pure 3:2, in centsone fifth carries the whole error Pitch and tuning

The wolf at the end of the chain

Put all the tuning error into one interval and eleven others become perfect. The twelfth becomes a howl, and for two centuries keyboards were built that way on purpose.

 CaGeDbAf♯Ec♯Ba♭F♯e♭C♯b♭A♭fE♭cB♭gFdone step= one fifth= one sharpouter ring: major keys · inner ring: their relative minors Scales and modes

Keys are neighbours, and the map is computed

Two keys a fifth apart share six of their seven notes. That single fact generates the circle of fifths, the key signatures, the shape of nearly every modulation, and the sense that some keys are far away.

 ICiiDiiiEIVFVGviAvii°BI – vi – IV – Vradius = semitones of voice motion from the tonic chord Harmony and voice leading

A progression is a path, and the map can be drawn

Chords in a key are not a list. They sit at measurable distances from home, and a progression is a walk across that space — which explains why some sequences feel like journeys and others like wandering.

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